Let $f$ be real-valued and differentiable at a point $c$ in $\mathbb R^n$, and assume that $||\nabla f(c)|| \neq 0.$ Prove that there is one and only one unit vector $u$ in $R^n$ such that $|f(c; u)|$ = $||\nabla f(c)||$ and that this is the unit vector for which $|f(c; u)|$ has its maximum value.
I can't think of any way to solve this, any hints are appreciated.
$f(c;u)$ is directional derivative of $f$ at $c$ in vector $u$'s direction